<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
a	O
channel	B-Application
system	I-Application
is	O
a	O
finite	B-Architecture
state	I-Architecture
machine	I-Architecture
similar	O
to	O
communicating	B-Application
finite-state	I-Application
machine	I-Application
in	O
which	O
there	O
is	O
a	O
single	O
system	O
communicating	O
with	O
itself	O
instead	O
of	O
many	O
systems	O
communicating	O
with	O
each	O
other	O
.	O
</s>
<s>
A	O
channel	B-Application
system	I-Application
is	O
similar	O
to	O
a	O
pushdown	B-Application
automaton	I-Application
where	O
a	O
queue	O
is	O
used	O
instead	O
of	O
a	O
stack	O
.	O
</s>
<s>
Formally	O
,	O
a	O
channel	B-Application
system	I-Application
(	O
or	O
perfect	O
channel	B-Application
system	I-Application
)	O
is	O
defined	O
as	O
a	O
tuple	O
with	O
:	O
</s>
<s>
Depending	O
on	O
the	O
author	O
,	O
a	O
channel	B-Application
system	I-Application
may	O
have	O
no	O
initial	O
state	O
and	O
may	O
have	O
an	O
empty	O
alphabet	O
.	O
</s>
<s>
A	O
configuration	O
or	O
global	O
state	O
of	O
the	O
channel	B-Application
system	I-Application
is	O
a	O
tuple	O
belonging	O
to	O
.	O
</s>
<s>
Given	O
a	O
perfect	O
or	O
a	O
lossy	O
channel	B-Application
system	I-Application
,	O
multiple	O
languages	O
may	O
be	O
defined	O
.	O
</s>
<s>
Most	O
problem	O
related	O
to	O
perfect	O
channel	B-Application
system	I-Application
are	O
undecidable	O
.	O
</s>
<s>
This	O
is	O
due	O
to	O
the	O
fact	O
that	O
such	O
a	O
machine	O
may	O
simulates	O
the	O
run	O
of	O
a	O
Turing	B-Architecture
machine	I-Architecture
.	O
</s>
<s>
Given	O
a	O
Turing	B-Architecture
machine	I-Architecture
,	O
there	O
exists	O
a	O
perfect	O
channel	B-Application
system	I-Application
such	O
that	O
any	O
run	O
of	O
of	O
length	O
can	O
be	O
simulated	O
by	O
a	O
run	O
of	O
of	O
length	O
.	O
</s>
<s>
Intuitively	O
,	O
this	O
simulation	O
consists	O
simply	O
in	O
having	O
the	O
entire	O
tape	O
of	O
the	O
simulated	O
Turing	B-Architecture
machine	I-Architecture
in	O
a	O
channel	O
.	O
</s>
<s>
The	O
content	O
channel	O
is	O
then	O
entirely	O
read	O
and	O
immediately	O
rewritten	O
in	O
the	O
channel	O
,	O
with	O
one	O
exception	O
,	O
the	O
part	O
of	O
the	O
content	O
representing	O
the	O
head	O
of	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
is	O
changed	O
,	O
to	O
simulate	O
a	O
step	O
of	O
the	B-Architecture
Turing	I-Architecture
machine	I-Architecture
computation	O
.	O
</s>
<s>
Multiple	O
variants	O
of	O
channel	B-Application
systems	I-Application
have	O
been	O
introduced	O
.	O
</s>
<s>
The	O
two	O
variants	O
introduced	O
below	O
does	O
not	O
allow	O
to	O
simulate	O
a	O
Turing	B-Architecture
machine	I-Architecture
and	O
thus	O
allows	O
multiple	O
problem	O
of	O
interest	O
to	O
be	O
decidable	O
.	O
</s>
<s>
A	O
one-channel	O
machine	O
is	O
a	O
channel	B-Application
system	I-Application
using	O
a	O
single	O
channel	O
.	O
</s>
<s>
The	O
same	O
definition	O
also	O
applies	O
for	O
all	O
variants	O
of	O
channel	B-Application
system	I-Application
.	O
</s>
<s>
When	O
the	O
alphabet	O
of	O
a	O
channel	B-Application
system	I-Application
contains	O
a	O
single	O
message	O
,	O
then	O
each	O
channel	O
is	O
essentially	O
a	O
counter	O
.	O
</s>
<s>
It	O
follows	O
that	O
those	O
systems	O
are	O
essentially	O
Minsky	B-Application
machines	I-Application
.	O
</s>
<s>
This	O
same	O
definition	O
applies	O
for	O
all	O
variants	O
of	O
channel	B-Application
system	I-Application
.	O
</s>
<s>
A	O
completely	O
specified	O
protocol	O
(	O
CSP	O
)	O
is	O
defined	O
exactly	O
as	O
a	O
channel	B-Application
system	I-Application
.	O
</s>
<s>
A	O
lossy	O
channel	B-Application
system	I-Application
or	O
machine	O
capable	O
of	O
lossiness	O
error	O
is	O
an	O
extension	O
of	O
completely	O
specified	O
protocol	O
in	O
which	O
letters	O
may	O
disappear	O
anywhere	O
.	O
</s>
<s>
A	O
lossy	O
channel	B-Application
system	I-Application
admits	O
two	O
kinds	O
of	O
steps	O
.	O
</s>
<s>
A	O
machine	O
capable	O
of	O
insertion	O
of	O
error	O
is	O
an	O
extension	O
of	O
channel	B-Application
system	I-Application
in	O
which	O
letters	O
may	O
appear	O
anywhere	O
.	O
</s>
<s>
This	O
process	O
allow	O
to	O
encode	O
any	O
channel	B-Application
system	I-Application
into	O
a	O
machine	O
capable	O
of	O
duplication	O
error	O
,	O
while	O
forcing	O
it	O
not	O
to	O
have	O
errors	O
.	O
</s>
<s>
Since	O
channel	B-Application
systems	I-Application
can	O
simulate	O
machines	O
,	O
it	O
follows	O
that	O
machines	O
capable	O
of	O
duplication	O
error	O
can	O
simulate	O
Turing	B-Architecture
machine	I-Architecture
.	O
</s>
<s>
This	O
section	O
contain	O
a	O
list	O
of	O
problems	O
over	O
channel	B-Application
system	I-Application
,	O
and	O
their	O
decidability	O
of	O
complexity	O
over	O
variants	O
of	O
such	O
systems	O
.	O
</s>
<s>
The	O
termination	O
problem	O
consists	O
in	O
deciding	O
,	O
given	O
a	O
channel	B-Application
system	I-Application
and	O
an	O
initial	O
configuration	O
whether	O
all	O
runs	O
of	O
starting	O
at	O
are	O
finite	O
.	O
</s>
<s>
This	O
problem	O
is	O
undecidable	O
over	O
perfect	O
channel	B-Application
systems	I-Application
,	O
even	O
when	O
the	O
system	O
is	O
a	O
counter	O
machine	O
or	O
when	O
it	O
is	O
a	O
one-channel	O
machine	O
.	O
</s>
<s>
This	O
problem	O
is	O
decidable	O
but	O
nonprimitive	O
recursive	O
over	O
lossy	O
channel	B-Application
system	I-Application
.	O
</s>
<s>
The	O
reachability	O
problem	O
consists	O
in	O
deciding	O
,	O
given	O
a	O
channel	B-Application
system	I-Application
and	O
two	O
initial	O
configurations	O
and	O
whether	O
there	O
is	O
a	O
run	O
of	O
from	O
to	O
.	O
</s>
<s>
This	O
problem	O
is	O
undecidable	O
over	O
perfect	O
channel	B-Application
systems	I-Application
and	O
decidable	O
but	O
nonprimitive	O
recursive	O
over	O
lossy	O
channel	B-Application
system	I-Application
.	O
</s>
<s>
This	O
problem	O
is	O
decidable	O
over	O
lossy	O
channel	B-Application
system	I-Application
and	O
trivially	O
decidable	O
over	O
machine	O
capable	O
of	O
insertion	O
of	O
errors	O
.	O
</s>
<s>
This	O
problem	O
is	O
undecidable	O
over	O
lossy	O
channel	B-Application
system	I-Application
.	O
</s>
<s>
The	O
recurrent	O
state	O
problem	O
consists	O
in	O
deciding	O
,	O
given	O
a	O
channel	B-Application
system	I-Application
and	O
an	O
initial	O
configuration	O
and	O
a	O
state	O
whether	O
there	O
exists	O
a	O
run	O
of	O
,	O
starting	O
at	O
,	O
going	O
infinitely	O
often	O
through	O
state	O
.	O
</s>
<s>
This	O
problem	O
is	O
undecidable	O
over	O
lossy	O
channel	B-Application
system	I-Application
,	O
even	O
with	O
a	O
single	O
channel	O
.	O
</s>
<s>
Given	O
a	O
system	O
,	O
there	O
is	O
no	O
algorithm	O
which	O
computes	O
a	O
finite	B-Architecture
state	I-Architecture
machine	I-Architecture
representing	O
for	O
the	O
class	O
of	O
lossy	O
channel	B-Application
system	I-Application
.	O
</s>
<s>
The	O
eventuality	O
property	O
,	O
or	O
inevitability	O
property	O
problem	O
consists	O
in	O
deciding	O
,	O
given	O
a	O
channel	B-Application
system	I-Application
and	O
a	O
set	O
of	O
configurations	O
whether	O
all	O
run	O
of	O
starting	O
at	O
goes	O
through	O
a	O
configuration	O
of	O
.	O
</s>
<s>
This	O
problem	O
is	O
undecidable	O
for	O
lossy	O
channel	B-Application
system	I-Application
with	O
impartiality	O
and	O
with	O
the	O
two	O
other	O
fairness	O
constraints	O
.	O
</s>
<s>
The	O
structural	O
termination	O
problem	O
consists	O
in	O
deciding	O
,	O
given	O
a	O
channel	B-Application
system	I-Application
if	O
the	O
termination	O
problem	O
holds	O
for	O
for	O
every	O
initial	O
configuration	O
.	O
</s>
<s>
Hierarchical	O
state	B-Architecture
machines	I-Architecture
are	O
finite	B-Architecture
state	I-Architecture
machines	I-Architecture
whose	O
states	O
themselves	O
can	O
be	O
other	O
machines	O
.	O
</s>
<s>
Since	O
a	O
communicating	O
finite	B-Architecture
state	I-Architecture
machine	I-Architecture
is	O
characterized	O
by	O
concurrency	O
,	O
the	O
most	O
notable	O
trait	O
in	O
a	O
communicating	O
hierarchical	O
state	B-Architecture
machine	I-Architecture
is	O
the	O
coexistence	O
of	O
hierarchy	O
and	O
concurrency	O
.	O
</s>
